Calculus Students’ Understanding of Volume in Non-Calculus Contexts

1. Calculus Students’ Understanding of Volume in Non-Calculus Contexts Allison Dorko RiSE Center University of Maine

2. Introduction & Rationale0 Calculus in general • Calculus courses are essential to math and science and are often prerequisites for undergraduates in STEM fields. • Calculus students often do not do as well as instructors might like (CBMS, 2000; Jencks & Phillips, 2001; Bressoud, 2005). • In many topics, calculus students are procedurally competent but lack a rich conceptual understanding (Ferrini-Mundy & Gaudard 1992; Ferrini-Mundy & Graham 1994; Milovanović 2011; Orton 1983; Rasslan & Tall 1997; Rosken 2007; Thompson & Silverman 2008). • The lack of a rich conceptual understanding creates issues later

3. Introduction & Rationale1 Understanding of volume • Volume plays a central role in calculus • Optimization; volumes of solids of revolution; etc. • These topics use derivatives and integrals, two topics with which calculus students struggle (Zandieh, 2000; Orton, 1983). • Volumes of solids of revolution are one of the most difficult topics for students (Orton, 1983) but it is not known what makes this so difficult. • Studies have found that elementary school students struggle with volume understanding, often finding surface area instead of volume (Battista & Clements, 1998). • There exists anecdotal evidence from calculus instructors that some calculus students also have issues with surface area and volume.

4. Optimization & Volume of Solid of Revolution

5. Introduction & Rationale3 Prior knowledge and new knowledge • Prior understanding may affect learning of new calculus concepts • Function (Carlsen, 1998; Monk, 1987) • Variable (Trigueros & Ursini, 2003) • The premise for my study is that something similar may be occurring with student understanding of volume and the understanding of calculus topics. • Knowing more about what issues calculus students have with volume may improve the teaching of volumes of solids of revolution, multiple integration, etc.

6. Research Questions • How successful are calculus students at finding volume? • Do calculus students find surface area when directed to find volume? • If calculus students find surface area when directed to find volume, what thinking leads them to do so? (yes)

7. Advanced Organizer • Research about students’ understanding of volume • Research design • Findings - Two stories: • 1. Finding surface area instead of volume • 2. Continuum of amalgam formulae

8. Literature: Elementary School Students’ Understanding of Volume • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae • We don’t know much about how calculus students understand volume • We do know something about how elementary school students understand volume. • Battista & Clements (1998) • Students counted cubes on the faces, often double-counting edges/corners • Specifically, some elementary school students find surface area when directed to find volume.

9. Research Design • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae • Cognitivist framework (Byrnes 2001; Siegler 2003) • Participants: 198 calculus I students • Data collection: two phases • Written surveys • Task-based clinical interviews • Data analysis • Grounded Theory (Strauss & Corbin, 1990) • Made use of other researchers’ findings about area and volume (e.g., Battista & Clements, 2003;Izsák 2005; Lehrer, 1998; Lehrer, 2003) and anecdotal evidence from calculus instructors

10. Research Instrument • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae

11. Categories of Student Responses • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae

12. Found Surface Area: Triangular Prism

13. Findings • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae

14. What thinking leads students to find surface area? • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae • Reason 1: Some students think that adding the areas of the faces of an object finds the measure of the object’s volume. • Reason 2: Some students understand the difference between area and volume, but mix the formulae together. • I call this mixed formula (e.g., V=2πr2h) an amalgam Volume of cylinder = πr2h SA of cylinder = 2πr2 + 2πrh Area of circle = πr2

15. Continuum of Amalgam Formulae • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae • Students used a myriad of formulae to find volume. I sorted these based on their surface area and volume elements. • Students’ volume formulae are useful for diagnosing their ideas/conceptions about volume

16. Answers to Research Questions • Literature Review • Research design • Findings • Surface area • Continuum of amalgam formulae • (1) How successful are calculus students at computational volume problems? • Somewhat successful; shape-dependent • (2) Do calculus students find surface area when directed to find volume? • Yes, approximately at the same rate as elementary school students • (3) If calculus students find surface area when directed to find volume, what is the thinking that leads them to do so? • Reason 1: Some students think that adding the areas of the faces of an object finds the measure of the object’s volume. • Reason 2: Some students understand the difference between area and volume, but mix the formulae together. • Difficulties calculus students have with volume-finding are similar to difficulties elementary school students have. • Viewing student formulae on a continuum of 2D, and 3D elements may help us diagnose their ideas about volume (and then design appropriate instruction)

17. Instructional Implications and Suggestions for Further Research • Instructional implications: create opportunities for students… • To revisit and strengthen their understanding of prerequisite topics in conjunction with the study of new content • To deepen their understanding of the connections between the dimensions of objects, the formulae for measurements of them, and the units associated with various spatial measures • Suggestions for further research: Does the Surface Area -Volume Amalgam interact with these students’ understanding of calculus topics that make use of these concepts? • Optimization • Volumes of Solids of Revolution

18. Works Cited • Battista, M.T., & Clements, D.H. (1998). Students’ understandings of three-dimensional cube arrays: Findings from a research and curriculum development project. n R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 227-248). Mahwah, NJ: Erlbaum. • Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics • Monk, G.S. (1989). A framework for describing student understanding of functions. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA. • Oehertman, M., Carlson, M., & Thompson, P. W. (2008). Foundational Reasoning Abilities that Promote Coherence in Students’ Function Understanding. In M. Carlson & C. Rasmussen (Eds.). Making the connection: Research and teaching in undergraduate mathematics education. (pp.27-41). MAA Notes #73. Mathematical Association of America: 2008. • Orton, A. (1983a). Students’ understanding of differentiation. Educational Studies in Mathematics, 15, 235-250. • Siegler, R. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, K., Marting, G., and Schifter, D. (Eds.) A Research Companion to Principles and Standards for School Mathematics. P. 289-303. Reston, VA: National Council of Teachers of Mathematics. • Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.

19. For more information… Email: allison.dorko@umit.maine.edu Reading material: Calculus Students’ Understanding of Area and Volume in Non-Calculus Contexts (masters thesis; Dec. 2011)